It is easily seen that the trajectory of the ODE stays in \(\mathcal {O}_p\) , and that \(f(X(t))\)  decreases with \(t\) .
Further assumptions on \(f\) , like Polyak-Lojasiewicz inequalities [1]}, [2]} or geodesic strong-convexity allow to prove the convergence of \(X(t)\)  to a minimizer as \(t\rightarrow + \infty \) .
If \(f\)  is Lipschitz then we have global convergence to a stationary point: \(\lim \inf \Vert \operatorname{Grad}f(X(t))\Vert  = 0\) .
In order to obtain a practical optimization algorithm, one should discretize the gradient flow.
Sadly, a naive Euler discretization, iterating \(X_{k+1}= X_k - \eta \operatorname{Grad}f(X_k)\)  with \(\eta >0\)  yields iterates which do not belong to the manifold, because the curvature is not considered.
